Download Optical Fibers and Waveguides

CHAPTER SEVENTEEN
Optical Fibers and Waveguides
17
Fiber and Integrated Optics
In this chapter we shall discuss from both a ray and wave standpoint
how light can be guided along by planar and cylindrical dielectric waveguides. We shall explain why optical fibers are important and useful in
optical communication systems and discuss briefly how these fibers are
manufactured. Some practical details of how fibers are used and how
they are integrated with other important optical components will conclude the chapter.
17.1 Introduction
We saw in the previous chapter that a Gaussian beam can propagate
without beam expansion in an optical medium whose refractive index
varies in an appropriate manner in the radial direction. This is a rather
specific example of how an optical medium can guide light energy. However, we can discuss this phenomenon in more general terms. By specifying the spatial variation in the refractive index, and through the use of
the wave equation with appropriate boundary conditions, we can show
that dielectric waveguides will support certain “modes” of propagation.
However, it is helpful initially to see what can be learned about this
phenomenon from ray optics.
Ray Theory of a Step-index Fiber
549
Fig. 17.1. Meridional ray entering fiber and being
guided by total internal reflection
Fig. 17.2. Skew rays
17.2 Ray Theory of a Step-index Fiber
A step-index fiber has a central core of index n1 surrounded by cladding
of index n2 where n2 < n. When a ray of light enters such a optical
fiber, as shown in Fig. (17.1), it will be guided along the core of the fiber
if the angle of incidence between core and cladding is greater than the
critical angle. Two distinct types of ray can travel along inside the fiber
in this way: meridional rays travel in a plane that contains the fiber
axis, skew rays travel in a non-planar zig-zag path and never cross the
fiber axis, as illustrated in Fig. (17.2).
For the meridional ray in Fig. (17.1), total internal reflection (TIR)
occurs within the core if θi > θc , or sin θi > n2 /n1 . From Snell’s law,
applied to the ray entering the fiber
sin θ = sin θo /n1 ,
(17.1)
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Optical Fibers and Waveguides
Fig. 17.3. Geometry for focusing light into fiber.
and since θ + θi = 90◦ , the condition for total internal reflection is
q
sin θ0 < n21 − n22 .
(17.2)
For most optical fibers the relative difference between the index of core
and cladding is ∆ = (n1 − n2 )/n1 is small, so Eq. (17.2) can be written
p
sin θ0 < (n1 − n2 )(n1 + n2 ),
(17.3)
which since n1 ' n2 can be written
√
sin θo < n1 2∆.
(17.4)
If a lens is used to focus light from a point source into a fiber, as shown
in Fig. (17.3), then there is a maximum aperture size D which can be
used. When the end of the fiber is a distance d from the lens, light rays
outside the cross-hatched region enter the fiber at angles too great to
allow total internal reflection. The quantity sin θo ' D/2d is called the
numerical aperture N.A. of the system. So, from Eq. (17.4)
N.A. = sin θo =
q
√
n21 − n22 = n1 2∆
(17.5)
If the point source in Fig. (17.3) is placed a great distance from the lens,
then d = f . In this case
D
N.A. =
.
(17.6)
2f
If the lens is chosen to be no larger than necessary, then the lens diameter
will be D. The ratio f /D is a measure of the focusing/light collecting
properties of the lens, called the f /number. So to match a distant source
to the fiber 2N.A. = 1/(f /number).
The Dielectric Slab Guide
551
Fig. 17.4. P -wave in a dielectric slab waveguide.
17.3 The Dielectric Slab Guide
Before considering the wave theory of a cylindrical fiber in detail we can
gain some insight into the propagation characteristics of meridional rays
by analyzing the two-dimensional problem of a dielectric slab waveguide.
A dielectric slab of dielectric constant 1 will guide rays of light by
total internal reflection provided the medium in which it is immersed
has dielectric constant 2 < 1 . For light rays making angle θ with the
axis as shown
in Fig. (17.4) total internal reflection will occur provided
q
sin θi > 21 .
Although there might appear to be an infinite number of such rays,
this is not so. As the ray makes its zig-zag path down the guide the
associated wavefronts must remain in phase or the wave amplitude will
decay because of destructive interference. We can analyze the slab as
a Fabry-Perot resonator. Only those rays that satisfy the condition for
maximum stored energy in the slab will propagate. This corresponds to
the upward and downward components of the travelling wave constructively interfering. The component of the propagation constant perpendicular to the fiber axis for the ray shown in Fig. (17.1) is k1 cos θi , so
for constructive interference
4dk1 cos θi = 2mπ,
which gives the condition
mλ0
.
(17.7)
4n1 d
For simplicity we have neglected the phase shift that occurs when the
wave totally internally reflects at the upper and lower core/cladding
boundaries.
A given value of m in Eq. (17.1) corresponds to a phase shift of 2mπ
cos θi =
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Optical Fibers and Waveguides
per round trip between the upper and lower boundaries of the slab: Such
a mode will not propagate if θ is greater than the critical angle. In other
words the cut-off condition for the mth mode is
d
m
= p 2
(17.8)
λo
4 n1 − n22
This condition is identical for waves polarized in the xz plane (P-waves)
or polarized in the y direction (S-waves). Only the lowest mode, m = 0,
has no cutoff frequency. A dielectric waveguide that is designed to allow the propagation of only the lowest mode is called a single-mode
waveguide. The thickness of the guiding layer needs to be quite small to
accomplish this. For example: at an operating wavelength of 1.55 µm
and indices n1 = 1.5, n2 = 1.485 the maximum guide width that will
support single-mode operation is 3.66 µm. Such waveguides are important because, in a simple sense, there is only one possible ray path in the
guide. A short pulse of light injected into such a guide will emerge as a
single pulse at its far end. In a multimode guide, a single pulse can travel
along more than one path, and can emerge as multiple pulses. This is
undesirable in a digital optical communication link. We can estimate
when this would become a problem by using Eq. (17.7)
For the m = 0 mode in a guide of total length L the path length is L.
For the m = 1 mode, the total path length along the guide is L + ∆L,
where
1
L + ∆L = L/(1 − (λo /4n1 d)2 ) 2
(17.8a)
A single, short optical pulse injected simultaneously into these two
modes will emerge from the fiber as two pulses separated in time by
an interval ∆τ where
n1 L
n1 L
n1 ∆L
∆τ =
=
(17.8b)
1 − c
co
o
2
co (1 − (λo /4n1 d) ) 2
This effect is called group delay. For example, if L = 1km, n1 = 1.5,
and d/λo = 5 then ∆τ = 2.78ns. Clearly, communication rates in excess
of about 200MHz would be impossible in this case. As the number of
modes increases, the bandwidth of the system decreases further.
17.4 The Goos-Hänchen Shift
We can examine the discussion of the previous section in a little more
detail by including the phase shift that occurs when a wave undergoes
total internal reflection. For a S-wave striking the core/cladding bound-
The Goos-Hänchen Shift
553
ary the reflection coefficient is
n1 cos θ1 − n2 cos θ2
ρ=
,
(17.a)
n1 cos θ1 + n2 cos θ2
where we have made use of the effective impedance for S-waves at angle
of incidence, θ1 , or refraction, θ2 , at the boundary, namely
Z0
0
=
,
Zcore
n1 cos θ1
Z0
0
Zcladding
=
(17.b)
n2 cos θ2
For angles of incidence greater than the critical angle, cos θ2 becomes
imaginary and can be written as
" #1/2
2
n1
2
cos θ2 = i
sin θ1 − 1
.
(17.c)
n2
In terms of the critical angle, θc ,
n1
cos θ2 = i (sin2 θ1 − sin2 θc )1/2
n2
n1
=i X
(17.d)
n2
If we write the phase shift on reflection for the S-wave as φs , then from
Eqs. (17.a) and (17.d)
cos θ1 − iX
ρ = |ρ|eiφs =
.
(17.e)
cos θ1 + iX
It is easy to show that |ρ| = 1: all the wave energy is reflected as a result
of total internal reflection. Therefore, if we write
cos θ1 − iX = eiφs /2 ,
then
(sin2 θ1 − sin2 θc )1/2
.
cos θ1
For P -waves the phase shift on reflection, φp , satisfies
tan(φs /2) = −
tan(φp /2) = −
(sin2 θ1 − sin2 θc )1/2
n2
n1 cos θ1
(17.f )
(17.g)
In practical optical waveguides usually n1 ' n2 so φs ' φp = φ. We can
put Eq. (17.f) in a more convenient form by using angles measured with
respect to the fiber axis. We define
θz = π/2 − θ1
θa = π/2 − θc
In a fiber with n1 ' n2 , both θz and θa will be small angles. With these
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Optical Fibers and Waveguides
definitions
(θa2 − θz2 )1/2
,
(17.h)
θz
If we represent the electric field of the guided wave in the slab as
φ = −2
Ei (z) = E1 eiβ1 z
then after total internal reflection the wave becomes
Er (z) = E1 e(β1 z+φ(θ1 ))
where β1 = k1 sin θ1 is the propagation constant parallel to the z axis
of the slab. Because this wave is laterally confined within the core we
can include the effects of diffraction by considering the wave as a group
of rays near the value θ1 . The phase shift φ(β1 ) can be expanded as a
Taylor series for propagation constants near β1 :
∂φ
φ(β) = φ(β1 ) + (β − β1 )
(17.i)
∂β β1
Therefore, the reflected wave can be written as
"
#
∂φ
Er (z) = E1 exp i(βz + φ(β1 ) + (β − β1 )
∂β β1
"
#
∂φ
= E0 exp i(βz + β
∂β β1
(17.j)
where all the terms independent of angle have been incorporated
into
the new complex amplitude E0 . The additional phase factor β ∂φ
∂β
β1
can be interpreted as a shift, Zs , in the axial position of the wave as it
reflects at the boundary, as shown in Fig. (17.5).
This shift in axial position Zs is called the Goos-Hänchen shift [A.L.
Snyder and J.D. Love, “The Goos-Hänchen shift,” Appl. Opt., 15, 236-8,
1976]
∂φ
Zs = −
(17.k)
∂β β1
which can be evaluated as
Zs = −
∂φ ∂θz
∂θz ∂β
(17.l)
β1
where
∂θz
1
=−
(17.m)
∂β
k sin θz
By differentiating Eq. (17.h) and assuming that the angle θz and θa are
Wave Theory of the Dielectric Slab Guide
555
small, we get
1/2
λ
1
(17.n)
πn1 θz (θa2 − θz2 )
This shift in axial position makes the distance travelled by the ray in
propagating a distance ` along the fiber shorter than it would be without
the shift. However, the shift is very small unless the ray angle is close to
the critical angle, in which case the evanescent portion of the associated
wave penetrates very far into the cladding.
Zs '
17.5 Wave Theory of the Dielectric Slab Guide
When a given mode propagates in a slab, light energy is guided along
in the slab medium, or core. There are, however, finite field amplitudes
that decay exponentially into the external medium. The propagating
waves in this case, because they are guided along by the slab, but are
not totally confined within it, are often called surface waves. This distinguishes them from the types of wave that propagate inside, and are
totally confined by, hollow conducting structures – such as rectangular
or cylindrical microwave waveguides. We can find the electromagnetic
field profiles in the one-dimensional slab guide shown in Fig. (17.4) by
using Maxwell’s equations. We look for a propagating solution with
fields that vary like eiωt−γz , where γ is the propagation constant.
There are both P and S-wave solutions to the wave equation. Clearly,
as can be seen from Fig. (17.4), the P wave has an Ez component of its
electric field so it could also be called a transverse magnetic field (TM)
wave. Similarly the S-wave is a transverse electric field (TE) wave. From
the curl equations, written in the form
curl E = −iωµH
(17.9)
curl H = iωE
the following equations result
∂Ez
(17.10a)
+ γEy = −iωµHx
∂y
∂Hz
= iωEy
∂x
∂Ez
−γEx −
= −iωµHy
∂x
∂Hz
+ γHy = iωEx
∂y
−γHx −
(17.10b)
(17.10c)
(17.10d)
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Optical Fibers and Waveguides
∂Ey
∂Ex
−
= −iωµHz
∂x
∂y
(17.10e)
∂Hy
∂Hx
(17.10f )
−
= iωEz
∂x
∂y
An expression for Hx can be found by eliminating Ey from Eqs. (17.10a)
and (17.10b), or vice-versa. Ex and Hy can be found from Eqs. (17.10c)
and (17.10d).
∂Hz
−1
∂Ez
Ex = 2
+ iωµ
)
(17.11)
(γ
γ + k2
∂x
∂y
Hy =
Ey =
γ2
−1
∂Ez
∂Hz
+γ
)
(iω
2
+k
∂x
∂y
1
∂Ez
∂Hz
(−γ
+ iωµ
)
γ 2 + k2
∂y
∂x
Hx =
γ2
1
∂Ez
∂Hz
(iω
+γ
)
2
+k
∂y
∂x
(17.12)
(17.13)
(17.14)
√
where k = ω µ.
From the Helmholtz equation (Eq. (16.4))
∇2 E + k 2 E = 0;
∇2 H + k 2 H = 0
(17.15)
Since ∂ /∂z is equivalent to multiplication by −γ Eq. (17.15) can be
written as
∂2E ∂2E
+
= −(γ 2 + k 2 )E
(17.16)
∂x2
∂y 2
2
2
2
∂2H ∂2H
+
= −(γ 2 + k 2 )H
∂x2
∂y 2
(17.17)
17.6 P-waves in the Slab Guide
For P -waves in the slab guide Ey = 0, ∂/∂y ≡ 0, so from Eq. (17.12)
−iω ∂Ez
Hy = − 2
(17.18)
γ + k 2 ∂x
and from Eq. (17.16)
∂ 2 Ez
= −(γ 2 + k 2 )Ez
(17.19)
∂x2
The solutions to Eq. (17.19) above are sine or cosine functions, or exponentials. If we wish the waves to be confined to the slab we must
choose the solution in medium 2 where the field amplitudes decay exponentially in the positive (or negative) x direction. So, for example,
P-waves in the Slab Guide
557
using Eqs. (17.18) and (17.19) the x dependence of Ez and Hy will be
iω2 −αx x
Ez2 = Be−αx x ; Hy2 = −
Be
x≥d
(17.20)
αx
Ez2 = Beαx x;
Hy2 =
iω2 αx x
Be
αx
Inside the slab the fields are
Ez1 = A sin kx x;
Hy1 =
x ≤ −d
−iω1
A cos kx x
kx
(17.21)
(17.22)
or
iω1
A sin kx x
kx
where we have taken note that in medium 2
Ez1 = A cos kx x;
Hy1 =
α2x = −(γ 2 + k22 )
(17.23)
(17.24)
where αx > 0, and in medium 1
kx2 = γ 2 + k12
(17.25)
where kx > 0.
A propagating wave exists in the slab provided γ is imaginary. If γ
were to be real then the e−γz variation of the fields would represent an
attenuated, non-propagating evanescent wave. For γ to be imaginary,
from Eq. (17.25), k12 > kx2 ; therefore, kx2 < ω 2 µ0 1 .
To find the relation between αx and kx we use the boundary condition
that at x = d the Ez and Hy fields (which are the tangential fields at
the boundary between the 2 media) must be continuous. Therefore,
Ez 1
Ez 2
=
(17.26)
Hy
Hy
1
x=d
2
x=d
and for the modes with odd Ez field symmetry in the slab, (the sine
solution above)
−αx
−kx
tan kx d =
,
(17.27)
iω1
iω2
so
2
αx = kx tan kx d.
(17.28)
1
It is straightforward to show that for the even symmetry Ez -field solution
in the slab, (the cosine solution above)
2
αx = − kx cot kx d.
(17.29)
1
To determine the propagation constant γ for the odd-symmetry modes
we must solve the simultaneous Eqs. (17.24), (17.25), and (17.28). Elimination of γ from Eqs. (17.24) and (17.25) gives
q
p
αx = k12 − k22 − kx2 = ω 2 µ0 (1 − 2 ) − kx2
(17.30)
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Optical Fibers and Waveguides
Fig. 17.5. Graphical solution of Eq. (17.31) for odd-symmetry
P -wave modes in a slab waveguide. Fig. (17.9) is similar.
We have assumed that µ1 = µ2 = µ0 , as most optically transparent
materials have relative permeability very close to unity.
From Eqs. (17.28) and (17.30):
s
1 ω 2 µ0 (1 − 2 )
tan kx d =
−1
(17.31)
2
kx2
The solution to this equation can be illustrated graphically, as shown in
Fig. (17.5). Whatever the value of ω there is always at least one solution
to this equation. For small values of kx d Eq. (17.28) gives
αx =
2 2
k d
1 x
(17.32)
and from Eq. (17.30)
1 2 2 2
1
) kx d = ( )2 ω 2 µ0 (1 − 2 )d2
2
2
4 4
neglecting the term in kx d gives
kx4 d4 + (
kx ' ω 2 µ0 (1 − 2 ) = k1 − k2
(17.33)
(17.34)
and from Eq. (17.25), if kx → 0
γ → ±ik1
(17.35)
Therefore as either 1 → 2 , or d becomes very small, αx also decreases
and energy extends far into medium 2. The propagation constant of the
wave approaches the value it would have in medium 2 alone.
The solution of Eq. (17.30) with the smallest value of kx d is called
the fundamental P -wave mode of the guide. This mode can propagate
at any frequency – it has no cut-off frequency. However, this is not true
for the even-symmetry Ez field solution. In this case, from Eqs. (17.29)
Dispersion Curves and Field Distribution
559
Fig. 17.6. Graphical Solution of Eq. (17.36) for even-symmetry
P -wave modes in a slab waveguide. Fig. (17.9) is similar.
and (17.30)
s
1
cot kx d = −
2
ω 2 µ0 (1 − 2 )
−1
kx2
(17.36)
The graphical solution of this equation is given in Fig. (17.6). In order
for this equation to have a solution, kx d must be greater than π/2. This
cannot be so while at the same time ω → 0.
17.7 Dispersion Curves and Field Distribution
For a propagating mode to exist, from either Eq. (17.31) or (17.36)
ω 2 µ0 (1 − 2 ) ≥ kx2 .
(17.37)
In this case Eq. (17.25) implies
ω 2 µ0 (1 − 2 ) ≥ γ 2 + ω 2 µ0 1 ,
(17.38)
which writing γ = ±iβ implies β ≥ ω µ0 2 . By solving Eq. (17.31) or
Eq. (17.36) for various values of ω we can plot the variation of β with ω
for the S and P wave modes. These curves are called dispersion curves.
From Eq. (17.25)
2
β 2 = k12 − kx2 = ω 2 µ0 1 − kx2
(17.39)
By solving for kx from either Eq. (17.31) for the odd symmetry Ez
solution or Eq. (17.36) for the even symmetry solution, we can produce
the dispersion curves shown in Fig. (17.7). The curve labelled m = 0 is
the fundamental mode, the curves labelled m = 1, 2, 3 etc. are alternate
even, odd, even symmetry Ez field higher modes. Some of their Ez field
distributions within the slab are shown in Fig. (17.8). The Ex and Hy
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Optical Fibers and Waveguides
Fig. 17.7.
Fig. 17.8. Ez field variation for P -wave odd and even-symmetry
modes in a slab waveguide. Fig. (17.10) is similar.
components of the P-wave solution are easily found from Eqs. (17.11)
and (17.12). For the fundamental mode,
Aγ
−γ ∂Ez
=−
cos kx x
2
+ k1 ∂x
kx
γ
∂Ez
Bγ −αx x
e
Ex (in medium 2) = − 2
=−
γ + k22 ∂x
αx
Ex (in medium 1) =
γ2
(17.40)
iω1 ∂Ez
iAω1
=−
cos kx x
γ 2 + k12 ∂z
kx
(17.41)
iω2 ∂Ez
−iBω2 −αx x
Hy (in medium 2) = − 2
=
e
γ + k22 ∂x
αx
The tangential component of magnetic field must be continuous at, say,
x = d. So from Eq. (17.41)
Hy (in medium 1) = −
A
2 kx e−αx d
=
B
1 αx cos kx d
(17.42)
S-waves in the Slab Guide
561
which from Eq. (17.28) gives
A
e−αx d
=
(17.43)
B
sin kx d
The actual value of A (or B) is determined from the energy flux in the
dielectric slab. The transverse intensity distribution in the guide is
I = Ex Hy ,
which from Eqs. (17.40) and (17.41) gives
iγw1 A2
I1 (in medium 1) =
cos2 kx x
kx2
iγw2 β 2 −2αx x
I2 (in medium 2) =
e
α2x
The total energy flux per unit width is
Z d
Z ∞
φ=2
I1 dx + 2
I2 dx
0
d
17.8 S-waves in the Slab Guide
For S-waves in the slab guide the electric field points only in the y
∂
≡ 0, and there are Hx and Hz components of magnetic
direction ∂y
field. In this case we start from Eq. (17.17) and write
∂ 2 Hz
= −(γ 2 + k 2 )Hz
(17.44)
∂x2
For the modes with odd symmetry of the Hz field
Hz1 = C sin kx x
| x |≤ d,
(17.45)
Cγkx
Cγ
cos kx x =
cos kx x,
(17.46)
Hx1 = 2
γ + k12
kx
iCωµ1
iCωµ1
Ey1 = 2
kx cos kx x =
cos kx x,
(17.47)
γ + k12
kx
Hz2 = De−αx x ,
x ≥ d,
(17.48)
Cγαx −αx x
Cγ −αx x
e
=
e
,
(17.49)
Hx2 = − 2
γ + k22
αx
iDωµ2 αx −αx x iDωµ2 −αx x
Ey2 = − 2
e
=
e
.
(17.50)
γ + k22
αx
It is straightforward to show, in a similar manner to before, that
µ2
αx =
kx tan kx d
(17.51)
µ1
562
Optical Fibers and Waveguides
Fig. 17.9.
and
s
tan kxd
µ1
=
µ2
ω 2 µ0 (1 − 2 )
− 1.
kx2
(17.52)
Even as ω → 0 there is always at least one solution to the equation. This
is the fundamental, m = 0, S-wave mode of the guide. The graphical
solution of Eq. (17.52) in Fig. (17.9) illustrates this.
Modes which have even symmetry of the Hz field have field components
Hz1 = C cos kx x,
| x |≤ d,
(17.53)
Cγ
Hx1 = −
sin kx x,
(17.54)
kx
iCωµ1
Ey1 = −
sin kx x,
(17.55)
kx
Hz2 = De−αx x ,
x ≥ d,
(17.56)
Cγ −αx
Hx2 =
e
,
(17.57)
αx
iDωµ2 −αx x
Ey2 =
e
.
(17.58)
αx
The defining equation for the modes is
µ2
(17.59)
αx = − kx cot kx d
µ1
Because most optical materials have µ ' µ0 the cutoff frequencies of the
S-wave modes above the fundamental are slightly lower, by a factor 21 ,
then the corresponding P -wave modes. Fig. (17.10) shows the schematic
variation of some of the field components of the S-wave modes in a
dielectric slab.
Practical Slab Guide Geometries
563
Fig. 17.10.
Fig. 17.11.
17.9 Practical Slab Guide Geometries
Slab guides that are infinite in one-dimension are convenient mathematically but do not exist in practice. Practical slab guides utilize index
changes or discontinuities in both the x and y directions (for a guide intended to propagate waves in the z direction). Some examples are shown
in Fig. (17.11). Such structures can be fabricated by methods similar to
those used in integrated circuit manufacture. In each case the fields outside the guiding layer decay exponentially. The fields are broadly similar
to those discussed previously for the infinite slab guide. However, the
modes that exist are no longer pure P or S-wave modes, they possess
all components of E and H, although they can still be predominantly P
or S in character in some circumstances.
Slab guide geometries exist in most semiconductor diode lasers and
in many LEDs. Consequently it is possible to integrate the light source
with a slab waveguide and then incorporate active components such as
modulators, couplers and switches into the structure. The resultant in-
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Optical Fibers and Waveguides
tegrated optic package can serve as a complete transmitting module with
full potential for amplitude or phase modulation, or multiplexing. We
shall meet examples of these devices later, in Chapter 19. In order to
realize a practical guided-wave optical communication link over long distances slab waveguides are not practical. The information-carrying light
beam must be transmitted in an optical fiber – a cylindrical dielectric
waveguide.
17.10 Cylindrical Dielectric Waveguides
The ray description of the slab waveguide is useful for describing the path
of a meridional ray down a fiber. However, to obtain a more detailed
understanding of the propagation characteristics of the fiber we again
use Maxwell’s equations. In cylindrical coordinates the curl equation
(Eq. 17.9) is:
1 ∂Ez
∂Eφ
∂Er
∂Ez
curl E = [
−
]r̂ + [
−
]φ̂
r ∂φ
∂z
∂z
∂r
1 ∂(rEφ ) 1 ∂Er
+[
−
]ẑ = iωµH
(17.60)
r ∂r
r ∂φ
with a similar equation for curl H. Er , Eφ , Ez , respectively, are the
radial, azimuthal, and axial components of the electric field, with corresponding unit vectors r̂, φ̂, and ẑ. The Helmholtz equations (Eq. 17.5)
become:
1 ∂2E
∂ 2 E 1 ∂E
+
= −(γ 2 + k 2 )E
+
∂r2
r ∂r
r2 ∂φ2
∂ 2 H 1 ∂H
1 ∂2H
+
+
= −(γ 2 + k 2 )H
(17.61)
∂r2
r ∂r
r2 ∂φ2
where, as before, k 2 = ω 2 µ.
We start by examining the solution of Eq. (17.61) for the axial component of the field. We look for a separable solution of the form
Ez (r, φ) = R(r)Φ(φ).
iωt−γz
(17.62)
The z dependence of Ez is e
, as before, but this factor will be
omitted for simplicity. Substituting from (17.62) into (17.61) gives
R00 rR0
Φ00
+
+ kc2 r2 = −
(17.63)
r2
R
R
Φ
where primes denote differentiation and we have written kc2 = γ 2 + k 2 .
The only way for Eq. (17.63) to be satisfied, since the L.H.S. is only
a function of r, and the R.H.S. is only a function of φ, is for each side
Practical Slab Guide Geometries
565
Fig. 17.12.
to be independently equal to a constant. Therefore,
R00 rR0
r2
+
+ kc2 r2 = ν 2
R
R
which is usually written as
1 ∂R
ν2
R00 +
+ (kc2 − 2 )R = 0
r ∂r
r
and
Φ00
−
= ν2
Φ
The general solution of Eq. (17.66) is
Φ(φ) = A sin νφ + B cos νφ.
(17.64)
(17.65)
(17.66)
(17.67)
ν must be a positive or negative integer, or zero so that the function
Φ(φ) is single valued: that is Φ(φ + 2π) = Φ(φ).
Eq. (17.65) is called Bessel’s equation. Its solutions are Bessel functions, which are the cylindrical geometry equivalents of the real or complex exponential solutions we encountered in the slab guide.
If kc2 > 0 the general solution of Eq. (17.65) is
R(r) = CJν (kc r) + DYν (kc r)
(17.68)
where Jν (kc r) is the Bessel function of the first kind and Yν (kc v) is the
Bessel function of the second kind.* These are oscillatory functions of r,
as shown in Fig. (17.12).
If kc2 < 0 the general solution of Eq. (17.68) is
R(r) = CIν (|kc |r) + D0 Kν (|kc |r)
(17.69)
where I(|kc |r), Kν (|kc |r) are the modified Bessel functions of the first
* The Yν Bessel function of the second kind is also written as Nν by some authors.
566
Optical Fibers and Waveguides
Fig. 17.13.
Fig. 17.14.
and second kinds, respectively. These are monotonically increasing and
decreasing functions of r as shown in Fig. (17.13).
Before proceeding further we can make several physical observations
to help simplify our analysis.
(i)
We cannot use the equations above to solve for wave propagation
in a graded index fiber - a fiber where the dielectric constant varies
smoothly in the radial direction. In such fibers is a function of
r (the quadratic index fiber is an example) and the wave equation
analysis is much more complex. References XX and XXX can be
consulted for details.
(ii) We can use these equations to find the field components in a stepindex fiber, whose radial refractive index profile is as shown in
Fig. (17.14). We find a solution for the core, where the dielectric
constant = n21 and match this solution to an appropriate solution
for the cladding, where 2 = n22 .
(iii) In the core we expect an oscillatory form of the axial field to be ap-
Practical Slab Guide Geometries
567
propriate (as it was for the slab guide). Since the function Yν (kc r)
blows up as r → 0 we reject it as physically unreasonable. Only
the Jν (kc r) variation needs to be retained.
(iv) We are interested in fibers with n1 > n2 that guide energy. The
fields in the cladding must decay as we go further away from the
case. Since Iν (kc r) blows up as r → ∞ we reject it as physically
unreasonable. Only the Kν (kc r) variation needs to be retained.
(v) In the core
kc21 > 0 implies γ 2 + k12 > 0
and for a propagating solution γ ≡ iβ; therefore
k12 > β 2
while in the cladding kc22 < 0, and therefore γ 2 + k22 < 0 which implies
k22 < β 2 . In summary
k2 < β < k1
The closer β approaches k1 , the more fields are confined to the core of the
fiber, while for β approaching k2 , they penetrate far into the cladding.
When β becomes equal to k2 the fields are no longer guided by the core:
they penetrate completely into the cladding.
(vi) If kc2 > 0, then the axial fields in the cladding would no longer
be described by the Kv Bessel function, instead they would be
described by the Jv Bessel function. These oscillatory fields no
longer decay evanescently in the cladding and the wave is no longer
guided by the core. We can take kc2 = 0 as a demarcation point
between a wave being guided, or leaking from the core.
If we take the sin νφ azimuthal variation for the axial electric field then
in the core we have,
Ez1 = AJν (kc1 r) sin νφ
(r ≤ a),
and in the cladding
Ez2 = CKν (| kc2 | r) sin νφ
(r ≥ a),
(17.70),
where the subscripts 1 and 2 indicate core and cladding fields, respectively. These fields must be continuous at the boundary between core
and cladding so
Ez1 (r = a) = Ez2 (r = a)
Far into the cladding the evanescent nature of the fields becomes apparent, since for large values of r
exp(−|kc2 |r)
Kν (|kc2 |r) ∝
(17.71)
(|kc2 |r)1/2
568
Optical Fibers and Waveguides
The corresponding magnetic fields are
Hz = BJν (kc1 ) cos νφ
Hz = DKν (kc2 ) cos νφ
(r ≤ a)
(r ≥ a)
(17.72)
In this case, where we have assumed the sin(νφ) for the electric field,
we must have the cos(νφ) variation for the Hz field to allow matching
of the tangential fields (which include both z and φ components) at the
core/cladding boundary. We must allow for the possibility of both axial
electric and magnetic field components in the fiber: so-called hybrid
modes. However, if ν = 0 then Eq. (17.70) shows that Ez = 0: this is a
TE mode. If we had chosen the cos(νφ) variation for the axial electric
field, then with ν = 0 we would have a TM mode. Thus, TE and TM
modes only exist for ν = 0. To find the relationship between the field
components orthogonal to the fiber axis and the Ez and Hz components
we work from Eq. (17.60) and the corresponding equation for curl H.
Some tedious algebra will yieldthe results
i γ Ez
1 ∂H
Er = − 2
+ ωµ
(17.73)
kc i ∂r
r ∂φ
i γ 1 ∂Ez
∂Hz
Eφ = − 2
− ωµ
(17.74)
kc i r ∂φ
∂r
i γ ∂Hz
1 ∂Ez
Hr = − 2
− ω
(17.75)
kc i ∂r
r ∂φ
i γ 1 ∂Hz
∂Ez
Hφ = − 2
+ ω
(17.76)
kc i r ∂φ
∂r
Note that for a TE mode (Ez = 0), Er ∼ ∂Hz /∂φ so that if Hz has a
variation ∼ cos(νφ), where ν is a constant, then Er must vary as sin(νφ).
For a TM mode (Hz = 0), Hr ∼ ∂Ez /∂φ with similar consequences. The
full variation of the field in core and cladding is:
17.10.1 Core:
ur
) sin νφ
a
Aγ 0 ur
iωµ0 ν
ur
= [−
J ( )+B
Jν ( )] sin νφ
(u/a) ν a
(u/a)2 r
a
Aγ ν
ur
iωµ0 0 ur
= [−
Jν ( ) + B
J ( )] cos νφ
(u/a)2 r
a
(u/a) ν a
ur
= BJν ( ) cos νφ
a
iω1 ν
ur
Bγ 0 ur
= [A
Jν ( ) −
J ( )] cos νφ
(u/a)2 r
a
(u/a) ν a
Ez = AJν (
(17.77)
Er
(17.78)
Eφ
Hz
Hr
(17.79)
(17.80)
(17.81)
Practical Slab Guide Geometries
Hφ = [−A
iω1 0 ur
Bγ ν
ur
J ( )+
Jν ( )] sin νφ
(u/a) ν a
(u/a)2 r
a
569
(17.82)
where u = kc1 a = (k12 +γ 2 )1/2 a is called the normalized transverse phase
constant in the core region.
17.10.2 Cladding:
wr
) sin νφ
a
wr Cγ
iωµ0 ν
wr
=[
Kν0
−D
Kν ( )] sin νφ
(w/a)
a
(w/a)2 r
a
Cγ ν
wr
iωµ0 0 wr
Kν ( ) − D
K ( )] cos νφ
=[
(w/a)2 r
a
(w/a) ν a
wr
= DKν ( ) cos νφ
a
iω2 ν
wr
Dγ
wr
= [−C
Kν ( ) +
K 0 ( )] cos νφ
(w/a)2 r
a
(w/a) ν a
iω2 0 wr
wr
Dγ ν
= [C
Kν ( ) −
Kν ( )] sin νφ
2
(w/a)
a
(w/a) r
a
Ez = CKν (
(17.83)
Er
(17.84)
Eφ
Hz
Hr
Hφ
(17.85)
(17.86)
(17.87)
(17.88)
where w =| kc | a = [−(k22 + γ 2 )]1/2 a is called the normalized transverse
attenuation coefficient in the cladding. If w = 0, this determines the
cut-off condition for the particular field distribution.
17.10.3 Boundary Conditions:
To determine the relationship between the constants A, B, C and D in
Eqs. (17.77-17.88) we need to match appropriate tangential and radial
components of the fields at the boundary. If we neglect magnetic behavior and put µ1 = µ2 = µ0 , for the tangential fields:
Ez1 = Ez2
Eφ1 = Eφ2
Hz1 = Hz2
Hφ1 = Hφ2
(17.89)
and for the radial fields
1 Er1 = 2 Er2
Hr1 = Hr2
(17.90)
where all fields are evaluated at r = a, and the subscripts 1 and 2 refer
to core and cladding fields, respectively. It can be shown that use of
these relationship in conjunction with Eqs. (17.77-17.88) leads to the
570
Optical Fibers and Waveguides
condition
Jν0 (u)
Kν0 (w) 1 Jν0 (u)
Kν0 (w) +
+
uJν (u) wKν (w) 2 uJν (u) wKν (w)
1 2 1
1 1
= ν2 2 + 2
+ 2
2
u
w
2 u
w
(17.91)
Furthermore,
u2 + w2 = (k12 + γ 2 )a2 − (k22 + γ 2 )a2 = (k12 − k22 )a2
(17.92)
which, since k1 = 2πn1 /λ0 , k2 = 2πn2 /λ0 , gives
u2 + w2 = 2k12 a2 ∆.
(17.93)
∆ is the relative refractive index difference between core and cladding
(n1 − n2 )
(n2 − n2 )
.
(17.94)
∆= 1 2 2 '
2n1
n1
A fiber is frequently characterized by its V number where
V = (u2 + w2 )1/2 ,
which can be written as
2πa 2
2πa
V =
(n − n22 )1/2 =
(N.A.)
λ0 1
λ0
= kn1 a(2∆)1/2
(17.95)
(17.96)
V is also called the normalized frequency.
Eqs. (17.91) and (17.93) can be solved for u and w to determine the
specific field variations in the fiber.
17.11 Modes and Field Patterns
The modes in a cylindrical dielectric waveguide are described as either
TE, TM or hybrid. The hybrid modes fall into 2 categories, HE modes
where the axial electric field Ez is significant compared to Er or Eφ , and
EH modes where the axial magnetic field Hz is significant compared to
Hr or Hφ . These modes are further characterized by two integers ν
and `. A mode characterized as HE12 , for example, has azimuthal field
variations like sin φ or cos φ, and since ` = 2, has 2 radial oscillations
of the axial electric field within the core. For the T E and T M modes
ν = 0. For the Ez field variation given by Eq. (17.77)
ur
Ez = AJν ( ) sin νφ
(17.97)
a
the number of radial oscillations of the axial electric field is determined
by how many times Jν (x) has crossed the axis as x varies from 0 to
ur/a. Each time the Bessel function oscillates from positive to negative,
The Weakly-Guiding Approximation
571
or vice-versa, there is a new value of (ur/a) where the oscillatory function
can merge smoothly with the radially decaying field distribution in the
cladding.
The field distributions are easiest to write down for the T E0` and
T N0` modes. Setting ν = 0 in Eqs. (17.77)-(17.52) we get the fields of
the T E0` mode, for example, in the core:
Ez = 0
Er = 0
iωµ0
ur
J1 ( )
(17.98)
(u/a)
a
ur
Hz = BJ0 ( )
a
Bγ
ur
Hr =
J1 ( )
(u/a)
a
We have made use of the relation J00 (x) = −J1 (x). The T E0` modes
correspond to choosing the constants A = C = 0 in Eqs. (17.77) (17.88).
We could have chosen the cos νφ variation for the Ez field in which
we would have got the fields of the T M0` mode by setting ν = 0, which
in this case implies B = D = 0.
The transverse fields of the mode give use to a characteristic Poynting
vector or intensity distribution, which can be called the mode pattern.
For the T E0` mode this is
Eφ = −B
ur
)
(17.99)
a
Because J1 (0) = 0, this is a donut-shaped intensity distribution as shown
in Fig. (17. ). The intensity distribution for the T M0` mode looks the
same. We shall see shortly that these are not the fundamental modes of
the step-index fiber. These modes cannot propagate at a given frequency
for an arbitrarily small core diameter—they have a cut-off frequency.
the only mode that has no cut-off frequency is the HE11 mode. To
discuss this further, and in order to find a way of grouping modes with
similar mode patterns together we need to work in the weakly-guiding
approximation.
S(r) = Eφ Hr ∝ J12 (
17.12 The Weakly-Guiding Approximation
In the weakly-guiding approximation we neglect the difference in refrac-
572
Optical Fibers and Waveguides
Fig. 17.15.
Fig. 17.16.
tive index between core and cladding and set 1 = 2 . In this case the
boundary condition relation, Eq. (17.91) becomes
Jν0 (u)
kν0 (w)
1
1
+
= ±ν
+
(17.100)
uJν (w) wkν (w)
u2 w 2
For the T E0` and T M0` mode this gives
J1 (u)
k1 (w)
+
=0
(17.101)
uJ0 (u) wK0 (w)
where we have made use of the relation K00 (w) = −K1 (w).
By the use of a series of relationships between Bessel functions, given
in Appendix , when the positive sign is taken on the right-hand side
of Eq. (17.100), the following equation results
Jν+1 (u) Kν+1 (w)
+
=0
(17.102)
uJν (n)
wKν (w)
This equation describes the EH modes. When the negative sign is taken
Mode Patterns
573
in Eq. (17.100) the following equation for the HE modes is obtained
Jν−1 (u) Kν−1 (w)
−
=0
(17.103)
uJν (u)
wKν (w)
By the use of the Bessel function relations given in Appendix
,
Eq. (17.109) can be rearranged into the form
Jν−1 (u)
Kν−1 (w)
+
(17.104)
uJν−2 (u) wKν−2 (w)
If we introduce a new integer m, which takes the values
(
1
for TE and TM modes,
m = ν + 1 for EH modes,
(17.105)
ν − 1 for HE modes,
then a unified boundary condition relation results that includes Eqs.
(17.101), (17.102) and (17.104). It is
uJm−1 (u)
wKm−1 (w)
=−
(17.106)
Jm (u)
Km (w)
If m − 1 < 0, as would be the case for HEν` modes with ν = 1, then
the Bessel function relations J−ν = (−1)ν Jν and K−ν = Kν would be
useful in evaluating Eq. (17.106).
If the V -number of the fiber is known, given by Eqs. (17.95) and
(17.96), then the values of u and w that satisfy Eq. (17.106) can be
determined. Once these parameters are known the fields in the fiber can
be calculated from Eqs. (17.77) - (17.88).
17.13 Mode Patterns
In the weakly-guiding approximation the Poynting vector variation
within the fiber is again calculated from
S = Et Ht ,
(17.107)
where the transverse fields in the fiber are
Et = Er r̂ + Eφ φ̂
Ht = Hr r̂ + Hφ φ̂.
(17.108)
The transverse intensity distribution has a characteristic mode pattern.
Similarities between these mode patterns allow groups of T E, T M and
hybrid modes to be grouped together as LP (linearly polarized) modes,
as shown in Fig. (17. ). Each LP mode is characterized by the two
integers m and `: ` describes the number of radial oscillations of the
field that occur in the core; m describes the number of variations of Ex
or Ey that occur as φ varies from 0 to 2π.
574
Optical Fibers and Waveguides
The schematic Poynting vector distributions for the LP modes shown
in Fig. (17. ) are the patterns appropriate to either (Ex Hy ) or (Ey Hx ).
For example, for the T E0` mode discussed earlier
Ex = −Eφ sin φ
Ey = Eφ cos φ
Hx = Hr cos φ
Hy = Hr sin φ
(17.109)
and the x-polarized intensity distribution can be written as
ur Ix ∝ J12
sin2 φ
(17.110)
a
For LPm` modes with ` ≥ 1 there are four constituent transverse or
hybrid modes. This results because each hybrid mode consists of two
degenerate modes, corresponding to the choice of the cos(νφ) or sin(νφ)
azimuthal variation of the fields. The T M or T E modes (for which
ν = 0) are singly degenerate and axisymmetric.
17.14 Cutoff Frequencies
The field in the cladding decays with increasing distance from the core
so long as w > 0. However, as w → 0, the propagation constant β
approaches the value k2 appropriate to the cladding and the wave is no
longer confined. The condition w = 0 determines the cutoff condition
for the mode in question, which from Eq. (17.106) becomes
uJm−1 (u)
=0
(17.111)
Jm (u)
The unique property of the HE11 mode can be illustrated with this
equation. This is the LP01 mode with m = 0.
Only the J0 (u) Bessel function is non-zero at u = 0, so Eq. (17.111)
has the solution u = 0 in this case. Therefore, the HE11 mode has no
cutoff frequency—or viewed another way, it can propagate in a fiber of
small core radius a. Since,
1/2
u = k12 + γ 2
a,
(17.112)
and at cutoff
u = V = k0 n1 a(2∆)1/2
(17.113)
u = 0 implies ω (the wave frequency) = 0. All other modes in the fiber
have a cutoff frequency, below which they cannot propagate. The cutoff
frequencies are determined by the zeros Jm−1,` of the Bessel function
Cutoff Frequencies
575
Table 17. . Low order zeros of Bessel functions.
J0
J1
J2
J3
J01 = 2.405
J02 = 5.520
J03 = 8.654
J11 = 3.832
J12 = 7.106
J13 = 10.173
J21 = 5.136
J22 = 8.417
J23 = 11.620
J31 = 6.380
J32 = 9.761
J33 = 13.015
Jm−1 , a few values of which are given in Table (17. ). For the T M0`
and T E0` modes the cutoff condition is
J0 (u)
=0
J1 (u)
(17.114)
so u = j0` at cutoff.
For the HEν` hybrid mode with ν ≥ 2 the cutoff condition is
Jν (u)
=0
(17.115)
Jν+1 (u)
so u = jν,` at cutoff.
For the EHν` hybrid modes the cutoff condition is
Jν−2 (u)
= 0,
(17.116
Jν−1 (u)
so u = jν−2,` at cutoff.
The smallest jm−1,` is j01 , so the LP11 has the lowest non-zero cutoff
frequency. The cutoff wavelength for this mode is found from
k0 n1 a(2∆)1/2 = 2.405
(17.117)
so the cutoff wavelength is
2πn1 a(2∆)1/2
(17.118)
2.405
If a given mode in the fiber is not below cutoff then its normalized phase
constant u has to take values that are bounded by values at which the
Jm−1 (u) and Jm (u) Bessel functions in Eq. (17.106) are equal to zero.
This is shown in Fig. (17. ) [Ref.: D. Gloge, “Weakly Guiding Fibers,”
Appl. Opt., 10, 2252-2258, 1971.]
The value of u must be between the zeros of Jm−1 (u) and Jm (u), that
is for modes with ` = 0, or 1.
λc =
Jm−1,` < u < Jm,`
If this condition is not met then it is not possible to match the J2 Bessel
function to the decaying Kν Bessel function at the core boundary. At
576
Optical Fibers and Waveguides
the lower limit of the permitted values of u in Fig. (17. ), the cutoff
frequency, V = u, while at the upper limit V → ∞.
17.14.1 Example:
For a fiber with a = 3µm, n1 = 1.5 and ∆ = 0.002 the cutoff wavelength
is λc = 744nm. This fiber would support only a single mode for λ > λc .
Alternatively, to design a fiber for single frequency operation at λ0 =
1.55µm we want
2.405λ0
a<
(17.119)
2πn1 (2∆)1/2
For n1 = 1.5 and ∆ = 0.002 this requires a < 6.254µm.
A fiber satisfying this condition would be a single-mode fiber for 1.55µ
m, it would not necessarily be a single-mode fiber for shorter wavelengths. Single-mode fibers for use in optical communication applications at the popular wavelengths of 810nm, 1.3µm or 1.55µm all have
small core diameters, although for mechanical strength and ease of handling their overall cladding diameter is typically 125µm. The fiber will
have additional protective layers of polymer, fiber, and even metal armoring (for rugged applications such as underground or overhead communications links, or for laying under the ocean). Fig. (17. ) shows the
construction of a typical multi fiber cable of this kind.
17.15 Multimode Fibers
If the core of a step-index fiber has a larger radius relative to the operating wavelength than the value given by Eq. (17.119) then more than
one mode can propagate in the fiber. The number of modes that can
propagate in a given fiber can be determined from its V -number. For
V < 2.405 the fiber will allow only the HE11 mode to propagate. For
relatively small V -numbers the number of modes that are permitted to
propagate can be determined by counting the number of Bessel function
zeros jν` < V . For large V -numbers a good estimate of the number M
of permitted modes is M = v 2 /2. [Ref. D. Gloge].
In a multimode fiber most of the modes will be well above cutoff
and well-confined to the core. The fraction of the mode power that
propagates in the cladding decreases markedly for these modes as is
shown in Fig. (17. ). [Fig. 5.11 from Gowar or Gloge Fig. 5.]
Multimode Fibers
577
17.16 Fabrication of Optical Fibers
There are two principal groups of fabrication techniques for the manufacture of low attenuation optical fibers: methods involving the direct
drawing of fiber from molten glass in a crucible; and various vapor deposition methods in which a “preform” with appropriate graded index
properties is produced and then drawn into a fiber. All these fabrication
procedures rely on the use of highly pure starting material whether this
be a borosilicate glass or silica, with added dopants. The brief discussion here can be supplemented with information from more specialized
texts [P.K. Cheo, “Fiber Optics and Optoelectronics,” Second Edition,
Prentice Hall, Englewood Cliffs, New Jersey, 1990, J.E. Midwinter, “Optical Fibers for Transmission, Wiley, New York, 1979; S.E. Miller and
A.G. Chynoweth, Eds., Optical Fiber Telecommunications, Academic,
New York, 1979; P.C. Schultz, “Progress in optical waveguide process
and materials,” Appl. Opt., 18, 3684-3693, 1979.]
Fig. (17. ) shows schematically the double crucible method for the
continuous production of clad fibers. The starting material can be fed
in in powder form, or as preformed glass rods. This technique cannot be
used for high melting point glasses, such as fused silica. The core and
cladding materials can be doped to give the core a higher index than
the cladding, for example, by doping with tantalum and sodium, respectively. By varying the drawing temperature the degree of interdiffusion
between core and cladding can be controlled to give a weakly-guiding
“step-index” or “graded-index” fiber. The addition of thallium to the
core, followed by its diffusion into the cladding during the drawing process also gives a graded index fiber. The double crucible method allows
the continuous production of fiber with attenuation as low as 5dB/km.
The production of single-mode versus multimode fiber is determined
by the size of the drawing aperture from the inner crucible, the temperature of the molten glass, and the rate at which the fiber is drawn.
For the production of the lowest attenuation fibers based on silica
doped with germanium, fluorine or phosphorus, particularly for the manufacture of single-mode fiber, a doped preform is fabricated by some form
of vapor deposition. The three principal vapor deposition processes are
shown schematically in Fig. 17. [D.B. Keck, “Single-mode fibers out
perform multimode cables,” IEEE Spectrum, March 1983, pp. 30-37].
In the outside chemical vapor deposition (OCVD) process materials
such as Sicl4 , gecl4 , POCl3 , BBr3 and, BCl3 are oxidized in a burner and
deposited on a rotating silica mandrel as a “soot”. By varying the vapor
578
Optical Fibers and Waveguides
to be decomposed, layers of soot particles can be built up, layer upon
layer, to generate the final refractive index profile desired. The mandrel,
with soot over layers is then melted into a preform in a furnace. The
preform can then be heated and drawn into a fiber.
In the inside chemical vapor deposition (ICVD) process the chemical
vapors are decomposed inside a silica tube by heating them externally
or with an internal RF-generated plasma. The tube with its inner layers
of soot can then be collapsed and inserted into a second silica (cladding)
tube to serve as the core of the final drawn fiber.
The vapor-phase axial deposition (VAD) process is a variant on the
OCVD process in which the core and cladding glasses are deposited simultaneously onto the end of the preform, which is continuously drawn
up into an electric furnace. The perform is rotated as deposition proceeds and as it is drawn up into the furnace its diameter decreases.
The final preform is then taken away and heated for fiber drawing [Ref.
P.K. Cheo].
17.17 Dispersion in Optical Fibers
The use of optical fibers to transmit data at ever increasing rates, which
in the laboratory have already exceeded 10 Gb/s over kilometer long
fibers, has required careful consideration of the effects of dispersion in
the fiber. In its simplest sense dispersion is the spreading in width of
a narrow optical pulse as it travels along the fiber. This spreading is
greatest in multimode fibers, where in a ray description each different
mode corresponds to a different length ray trajectory inside the fiber.
The different pulse transit times along these different ray paths leads to
broadening or splitting, of a narrow optical pulse. This is called modal
dispersion. For example, for a step-index fiber the transit time difference
between a ray that propagates along the axis and one that enters just
within the N.A. is easily shown to be
n2 `
∆T = 1 ∆
n2 c0
(17.120)
n1 `∆
'
c0
For ∆ = 0.002, n1 = 1.5, this modal dispersion is 10ns/km.
In single-mode fibers modal dispersion does not exist but dispersion
still occurs. This residual dispersion comes from material dispersion
dn/dλ, the variation of the refractive index of core and cladding with
Dispersion in Optical Fibers
579
wavelength (or frequency), and waveguide dispersion the variation of the
waveguide propagation constant β with frequency. A detailed analysis
of these effects has been given by Gowar, but a few general observations
can be made.
17.17.1 Material Dispersion
The transit time of a pulse in a single-mode fiber depends, among other
factors, on its group velocity,* defined as
dω
vg =
(17.121)
dβ
In the simplest case of a pulse totally confined to the core
ωn1
β = k1 =
c0
It is easy to show that
c0
vg =
(17.121)
1
(n1 − λ dn
dλ )
We can define a group refractive index, ng1 , where
c0
vg =
(17.122)
ng1
If a narrow optical pulse is travelling in the core, then its different spectral components will travel with different velocities. The spectral components covers a wavelength range ∆λ that depends on the monochromaticity of the source. For a light emitting diode (LED) ∆λ is typically
' 40 nm at 800nm; for a semiconductor laser the equivalent spectral
frequency width will be in the range 1-100MHz. The transit time of the
pulse will be
dn1 `
`
T =
= n1 − λ
,
(17.123)
vg1
dλ c0
and a narrow pulse (impulse) will speed to a width
` d2 n ∆λ
∆T = − λ2 2
.
(17.124)
c0 dλ
λ
This pulse broadening is minimized, for a source with specific spectral
purity ∆λ/λ when the quantity
d2 n
|Ym | = |λ2 2 |
(17.125)
dλ
* The group velocity is a measure of the velocity with which the centroid of a pulse,
which itself contains a range of frequencies, will travel. It is therefore, an important
parameter in characterizing the performance of an optical communication system
that transmits information as a train of light pulses.
580
Optical Fibers and Waveguides
is minimized. For doped silica fibers Ym = 0 for wavelengths in the
range 1.22µm < λ < 1.37µm, the minimum being at λmd = 1.276µm
for a pure silica fiber. The spreading of the pulse will not actually be
zero, even at this operating wavelength, because of the spectral width of
the source. The residual pulse broadening that occurs for a source with
λ0 = λmd can be shown to be
2
∆T
λ3 d3 n
∆λ
= (−) 0
`
8c0 dλ3 λ0
λ
For pure silica at λ0 = 1.276µm, ∆T /` = 2 × 10−11 (∆λ/λ)2 s/m.
For a high data rate optical communication link using a single mode
semiconductor laser running at 10 Gbit/s the effective laser linewidth
is broadened to ' 20 GHz, so ∆λ/λ = 0.00009 and ∆T /` ' 1.5 ×
10−16 s/km.
Operation at a wavelength of 1.55µm, where the attenuation of silica
is a minimum, but Ym = −0.01, leads to more broadening. Even so,
this broadening is extremely small. For the optical link described above
running at 1.55µm, ∆T /` ' 3.5ps/km.
17.17.2 Waveguide Dispersion
When a mode propagates, its propagation constant is γ = iβ where β
simultaneously satisfies both the equations
β = [k12 − (u/a)2 ]1/2
and
β = [(w/a)2 + k22 ]1/2
At cutoff w = 0, β = k2 and the wave propagates completely in the
cladding. As the frequency ω of the wave rises far above the cut-off
value β → k1 . The variation of β with ω is called a waveguide dispersion
curve. At any point on a dispersion curve the phase velocity of the mode
is ω/β, while the group velocity is dω/dβ. The dispersion curve for a
particular mode is bounded by the β values appropriate for core and
cladding as shown in Fig. (17. ). The so called light lines have ω/β
slopes c1 and c2 , where c1 and c2 are the velocity of light in core and
cladding, respectively.
17.18 Coupling Optical Sources and Detectors to Fibers
In order to couple light from a source into a fiber efficiently the light
Coupling Optical Sources and Detectors to Ribers
581
Fig. 17.17.
from the source must, in the first instance, be injected within the numerical apeature. In the case of multimode fibers this is generally sufficient
to ensure efficient collection and efficient guiding of the injected source
light. However, when step-index single-mode fibers are comenced the
injected light must additionally be matched as closely as possible to the
lateral intensity profile of the mode that will propagate in the fiber.
This cannot be done exactly because the injected light will typically
have a Gaussian radial intensity profile, whereas the single mode lateral
intensity profile is a zero-order Bessel function with exponentially decaying wings. For optimum coupling a circularly symmetric Gaussian
laser beam should be focused so that its beam waist is on, and parallel
to, the cleaned end face of the fiber, as shown in fig. (17.17).
Optimum coupling of the Gaussian beam into the HE11 mode occurs when w0 = 0.61a and the v-number of the fiber is V = 3.8,[1] in
otherwords close to the value where the next highest mode HE12 will
propagate. In this situation, if Fresnel losses at the fiber entrance face
are ignored, the maximum coupling efficiency is 85%. The coupling efficiency from a Gaussian beam to a single mode or multimode fiber is
somewhat complex to calculate in the general case[1] . To optimize coupling the input Gaussian must be matched to the equivalent Gaussian
profile of the propagating mode. A useful approximation that can be
used for a step-node fiber of core radius a is
a
wmode =
sqrt2 ln v
If the end of the fiber is illuminated with too large a focused spot significant energy enters higher order mode, which leak out of the cladding.
This is really demonstrated in the laboratory. If a visible laser is focused into a fiber the first few centimeters of the fiber will glow brightly
582
Optical Fibers and Waveguides
as non-propagating modes are excited and then energy leaks from the
cladding.
In the arrangement shown in Fig. (17. ) the lens can very conveniently
be replaced by a GRIN lens, which allow butt coupling of source to
GRIN. The parameters of the GRIN lens must be chosen comnensurate
with the input laser spot size to give an appropriate beam waist at the
output face of the GRIN.
If the laser source to be coupled to the fiber is not generating a circularly symmetric Gaussian beam, then anamorphic focussing optics must
be used to render the beam circular and focus it onto the end face of the
fiber. A typical semiconductor laser may generate an elliptical Gaussian
beam, one in which the beam is characterized by different spot sizes
in two orthogonal directions. In this case a combination of a cylindrical
and spherical lens can be used to give improved focusing. If the emitting
facet of the semiconductor laser is sufficiently small then a microfabricated spherical lens can be used to sample source and fiber as we have
seen in Chapter 13.
The problem of coupling an optical detector to a fiber efficiently is
an easy one. Most optical detectors are of larger area than the model
spot leaving the fiber so direct butt coupling from fiber to detector is
straightforward.
17.18.1 Fiber Connectors
If two identical fibers are to be joined together this can be done with
either mechanical coaxial connectors or through a fusion spheric. In
either case the flat, orthogonally cleared end of the two fibers must be
brought together so that their two faces are in very close proximity,
parallel, and the fibers are coaxial, as shown in Fib. (17.18). Any tilt or
offset of the fibers will reduce the coupling efficency.
Precision, mechanical connectors are able to couple two fibers with
typical losses of 0.1-0.2dB. Generally, the connector will contain a small
quantity of index-matching fluid that is squared in the small interface
gap between the 2 fibers.
In a fusion sphere the two fiber ends in Fig. (17. a) are brought into
contact and external heat quickly applied to melt and fuse the two fibers
together. Such spheres will have an insertion loss of 0.1dB.
17.19 Problems for Chapter 17
Problems for Chapter 17
583
Fig. 17.18. Alignment of fibers for connection.
(1)
(2)
The Goos-Hänchen shift can be regarded as equivalent to the penetration of the totally internally reflected ray into the cladding so
that the “effective” thickness of the slabs is 2deff , as shown below.
Derive an expression for deff using the parameters in Eq. (17.h).
Prove that for an optical fiber being operated at a center wavelength λ0 with a source of spectral width ∆λ the minimum pulse
broadening as a function of distance caused by material dispersion
is
2
∆T
λ3 d3 n
∆λ
= (−) 0
`
8c0 dλ3 λ0
λ
(3)
A fiber with a parabolic index profile has
1
n(r) = n0 − n2 r2
2
with n0 = 1.5 and n2 = 109 m−2 . Design a single lens focusing
arrangement that will optimally couple a laser with λ0 = 180nm
and w0 = 2mm into the fiber. How would you modify the focusing
arrangement if the laser emitted a Gaussian beam with w0 = 5µm
from the laser facet?
(4)
Repeat question (3) but design a focusing system using a parabolic
index GRIN lens that is 20mm long. What parabolic profile parameter n2 for the GRIN lens is needed?
REFERENCES
Snyder and Love.